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arXiv · 2404.18974

$\Pi^0_4$ conservation of Ramsey's theorem for pairs

Quentin Le Houérou·Ludovic Levy Patey·Keita Yokoyama
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Abstract

In this article, we prove that Ramsey's theorem for pairs and two colors is a $\forall \Pi^0_4$ conservative extension of $\mathsf{RCA}_0 + \mathsf{B}\Sigma^0_2$, where a $\forall \Pi^0_4$ formula consists of a universal quantifier over sets followed by a $\Pi^0_4$ formula. The proof is an improvement of a result by Patey and Yokoyama and a step towards the resolution of the longstanding question of the first-order part of Ramsey's theorem for pairs.

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Quentin Le Houérou, Ludovic Levy Patey, Keita Yokoyama. 2024-04-29. $\Pi^0_4$ conservation of Ramsey's theorem for pairs. https://arxiv.org/abs/2404.18974

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